{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "53ce4b3e-f901-4bde-aa79-f65c2cc47277",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-09-07T02:39:43.326014Z",
     "iopub.status.busy": "2024-09-07T02:39:43.325647Z",
     "iopub.status.idle": "2024-09-07T02:39:43.695812Z",
     "shell.execute_reply": "2024-09-07T02:39:43.694019Z",
     "shell.execute_reply.started": "2024-09-07T02:39:43.325989Z"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fed7d5ca-997e-4e54-a829-a7d64212be04",
   "metadata": {},
   "source": [
    "### 推荐《深度学习详解》"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a28e0c4b-5f5a-4a41-ae7a-a299ba3892fd",
   "metadata": {},
   "source": [
    "- 豆瓣 9.0\n",
    "    - https://book.douban.com/subject/36997460/\n",
    "    - github 配置资源：https://github.com/datawhalechina/leedl-tutorial\n",
    "- 系统性地概括了李宏毅的深度学习系列\n",
    "    - 概括和浓缩\n",
    "    - 李宏毅老师的课程都比较轻松，深入浅出，好入门；\n",
    "    - 结合视频，课前预习和课后复习\n",
    "- 深度学习基础理论的入门\n",
    "- 适合整体性地 review\n",
    "    - 比如面试前；"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "920ec5c5-ff31-4ce3-8fb6-bbcb4a77ddde",
   "metadata": {},
   "source": [
    "### why cross entropy loss"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d7609e81-4fe2-4dd1-8a0d-17b75e398ab6",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{split}\n",
    "&\\ell=\\sum_i(y_i-y_i')^2\\\\\n",
    "&\\ell=-\\sum_iy_i\\ln y_i'\\\\\n",
    "\\end{split}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "f58880ed-1ee9-46a2-90ae-fbc7963d19f2",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-09-07T02:34:51.033883Z",
     "iopub.status.busy": "2024-09-07T02:34:51.033532Z",
     "iopub.status.idle": "2024-09-07T02:34:51.041949Z",
     "shell.execute_reply": "2024-09-07T02:34:51.040091Z",
     "shell.execute_reply.started": "2024-09-07T02:34:51.033859Z"
    }
   },
   "outputs": [],
   "source": [
    "def softmax(x):\n",
    "    exp_x = np.exp(x - np.max(x))  # 为了数值稳定性，减去输入的最大值\n",
    "    return exp_x / np.sum(exp_x, axis=-1, keepdims=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "id": "bb7bab07-d0e9-4152-98cf-5234be290b6e",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-09-07T03:59:30.424146Z",
     "iopub.status.busy": "2024-09-07T03:59:30.423835Z",
     "iopub.status.idle": "2024-09-07T03:59:30.432613Z",
     "shell.execute_reply": "2024-09-07T03:59:30.430676Z",
     "shell.execute_reply.started": "2024-09-07T03:59:30.424124Z"
    }
   },
   "outputs": [],
   "source": [
    "def mse(gt, logits, epsilon=1e-7):\n",
    "    probs = softmax(logits)\n",
    "    # probs = np.clip(probs, epsilon, 1. - epsilon)\n",
    "    return np.sum((gt - probs)**2)\n",
    "\n",
    "def ce(gt, logits, epsilon=1e-7):\n",
    "    probs = softmax(logits)\n",
    "    probs = np.clip(probs, epsilon, 1. - epsilon)\n",
    "    return -np.sum(gt * np.log(probs))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "id": "0f422429-33b2-4dae-9ffc-0289661e4dd1",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-09-07T04:20:54.688284Z",
     "iopub.status.busy": "2024-09-07T04:20:54.687661Z",
     "iopub.status.idle": "2024-09-07T04:20:54.699259Z",
     "shell.execute_reply": "2024-09-07T04:20:54.696935Z",
     "shell.execute_reply.started": "2024-09-07T04:20:54.688236Z"
    }
   },
   "outputs": [],
   "source": [
    "y = np.asarray([1, 0, 0])\n",
    "logits = np.asarray([-5, 5, -1000])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "id": "a5ceec8f-b691-4727-ab6e-90149e32ecf2",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-09-07T04:20:56.003797Z",
     "iopub.status.busy": "2024-09-07T04:20:56.002377Z",
     "iopub.status.idle": "2024-09-07T04:20:56.015917Z",
     "shell.execute_reply": "2024-09-07T04:20:56.013849Z",
     "shell.execute_reply.started": "2024-09-07T04:20:56.003732Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([4.53978687e-05, 9.99954602e-01, 0.00000000e+00])"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "softmax(logits)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "id": "39a96e63-b984-4dfb-a2fd-911be4a6ac74",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-09-07T04:01:05.135249Z",
     "iopub.status.busy": "2024-09-07T04:01:05.134639Z",
     "iopub.status.idle": "2024-09-07T04:01:05.147391Z",
     "shell.execute_reply": "2024-09-07T04:01:05.145073Z",
     "shell.execute_reply.started": "2024-09-07T04:01:05.135202Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1.9998184126471235, 10.000045398899216)"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mse(y, logits), ce(y, logits)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "id": "089078bb-77ce-444e-bd49-93b723de8a14",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-09-07T03:57:50.904801Z",
     "iopub.status.busy": "2024-09-07T03:57:50.904542Z",
     "iopub.status.idle": "2024-09-07T03:57:51.750645Z",
     "shell.execute_reply": "2024-09-07T03:57:51.749824Z",
     "shell.execute_reply.started": "2024-09-07T03:57:50.904785Z"
    }
   },
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<Figure size 2000x800 with 4 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# 创建网格\n",
    "logits1 = np.linspace(-10, 10, 100)\n",
    "logits2 = np.linspace(-10, 10, 100)\n",
    "X, Y = np.meshgrid(logits1, logits2)\n",
    "\n",
    "# 计算MSE和CE\n",
    "Z_mse = np.zeros_like(X)\n",
    "Z_ce = np.zeros_like(X)\n",
    "\n",
    "for i in range(X.shape[0]):\n",
    "    for j in range(X.shape[1]):\n",
    "        logits = np.array([X[i,j], Y[i,j], -1000])\n",
    "        Z_mse[i,j] = mse(y, logits)\n",
    "        Z_ce[i,j] = ce(y, logits)\n",
    "\n",
    "# 绘制热力图\n",
    "fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(20,8))\n",
    "\n",
    "im1 = ax1.imshow(Z_mse, extent=[-10, 10, -10, 10], \n",
    "                 origin='lower', aspect='auto', \n",
    "                 # cmap='viridis'\n",
    "                 cmap='rainbow'\n",
    "                )\n",
    "ax1.set_title('MSE Loss')\n",
    "ax1.set_xlabel('logits1')\n",
    "ax1.set_ylabel('logits2')\n",
    "plt.colorbar(im1, ax=ax1)\n",
    "\n",
    "im2 = ax2.imshow(Z_ce, extent=[-10, 10, -10, 10], \n",
    "                 origin='lower', aspect='auto',  \n",
    "                 # cmap='viridis'\n",
    "                 cmap='rainbow'\n",
    "                )\n",
    "ax2.set_title('Cross-Entropy Loss')\n",
    "ax2.set_xlabel('logits1')\n",
    "ax2.set_ylabel('logits2')\n",
    "plt.colorbar(im2, ax=ax2)\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "id": "1987db42-6f2c-415b-a2b8-bd196a62f44f",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2024-09-07T03:59:56.742345Z",
     "iopub.status.busy": "2024-09-07T03:59:56.742022Z",
     "iopub.status.idle": "2024-09-07T03:59:56.752077Z",
     "shell.execute_reply": "2024-09-07T03:59:56.750574Z",
     "shell.execute_reply.started": "2024-09-07T03:59:56.742323Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.9999909587130262 13.000002260326852\n",
      "1.999996673889272 14.000000831528373\n"
     ]
    }
   ],
   "source": [
    "print(mse(y, [-5, 8, -1000]), ce(y, [-5, 8, -1000]))\n",
    "print(mse(y, [-5, 9, -1000]), ce(y, [-5, 9, -1000]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "aaa032c5-df45-4bb0-a15d-1680675f1e19",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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